Metrics on quantum channels: issues, fixes, applications

Seminar date and time: 
Wednesday, 8 March, 2017 - 12:00
Andreas Winter
GIQ - Seminar room

The diamond norm on superoperators (aka 'completely bounded norm' in the Heisenberg picture) provides a natural, operationally well-motivated, and computable metric on quantum channels (cptp maps). In particular, it quantifies the optimal bias in hypothesis testing between two channels, it is given by a semidefinite programme, and it enjoys a number of good mathematical properties. It furthermore provides the natural setting when discussing continuity of channel capacities. In  some situations, however, notably in infinite dimension, the  diamond norm is too strong to be reasonably applicable. This  can be illustrated with simple Bosonic channels. Inspired by communication theory, i propose a definition of  diamond norm with an energy constraint (with respect to a given  Hamiltonian), and show that this allows for a resolution of most of the issues of the diamond norm, while retaining its good properties. As an application, i show how to prove the continuity of Bosonic quantum channel capacities (C, Q, P, etc) with an energy constraint at the input.

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